Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity

Abstract

For 2D compressible isentropic Euler equations of polytropic gases, S. Alinhac establishes that the axisymmetric smooth solution blows up. In the present paper, for 2D compressible isentropic Euler equations of Chaplygin gases, we show that the small perturbed smooth solution exists globally when the rotationally invariant initial data are a perturbation of size ε>0 of a rest state. Near the light cone, 2D Euler equations of Chaplygin gases can be transformed into a second order quasilinear wave equation, which satisfies both the null conditions. This will lead to that the corresponding second order quasilinear wave equation admits a global smooth solution near the light cone. However, away from the light cone, the hydrodynamical waves have no decay in time and strongly affect the related acoustical waves. By introducing a nonlinear ODE, we can distinguish the fast decay part and non-decay part so that the global energy estimates are derived.